1Indexing X-ray powder diffraction patterns using the homology methodDo we always need to apply brute force programs for automatic indexing, or in many cases the problem will be solved with our crystal chemistry knowledge and a pocket calculator? Could we extract useful information from the intensities of the reflections, which will help us to index the pattern? Could we get an idea about the structure model while determining unit cell parameters? Lets try to use an elegant ancient technique, which people developed before the computer era. homologos - similar, conformable (Greek) 2Everybody can manually index XRD pattern of a cubic structureindexation is based on analyzing Q seriescubic lattice with a 10 Å 3Distortions of cubic lattice 4SublatticeLets assume that your new structure with (a, b, c, a, b, g) is a derivative from some more simple structure, which we will call the substructure with (asub, bsub, csub, asub, bsub, gsub)

Usually the substructure is a highly symmetric structure,

belonging to one of the simple prototypes (perovskite, rock

salt, fluorite, rutile, shpinel, pyrochlore etc.)

2. We will call a reflection a sublattice reflection if its h,k,l are related to hsub, ksub, lsub by a unit matrix with det1. All other reflections are the superlattice reflections. Note that the subcell symmetry can be lower than the true symmetry of the structure

13Indexing of the XRD powder pattern of Ca3ReO6 For the monoclinic unit cell with ac we always can find an orthorhombic unit cell of larger volume monoclinic P ? orthorhombic B 14Indexing of the XRD powder pattern of Ca3ReO6 Lattice parameters of the new B-centered orthorhombic unit cell a 5.5363(3)Å, b 3.9921(3)Å c 5.8022(4)Å What to do with other unindexed reflections? We will treat them as superlattice reflections. 15Superstructure1. Cation ordering2. Anion ordering3. Ordering of anion vacancies4. Ordering of cation vacancies5. Atomic displacements 16SuperstructureTetragonal altc1. Superstructure is related to increasing unit cell volume (determinant of the transformation matrix gt 1)2. Intensities of superlattice reflections depend strongly on the degree of deviation from the substructure. 3. Transformation matrix shows the relationships between the sublattice and superlattice vectors4. Transformation matrix does not allow you to calculate indexes of the superlattice reflections. 17Indexing of the XRD powder pattern of Ca3ReO6 Searching for the superstructure is a non-trivial step. Low angle reflections are the most important. Check possible multiplication of subcell axes lengths. Check different possibilities of simple hkl for low angle reflections. Check relationships between the low angle and sublattice reflections. In our case doubling of b-parameter Q010 628/4 157 Q011 157 1188/4 454 first low angle reflection Q110 157 1305/4 483 second low angle reflection Final refinement of the lattice parameters a 5.5366(3)Å, b 7.9845(4)Å c 5.8022(3)Å 18What do we get extra? Transformation matrixTransformation of the atomic coordinates - inverse transformation matrixYou immediately get a starting set of atomic coordinates in the superstructure unit cell for further work, for example, to use as a first model for the Rietveld refinement. You can limit the choice of possible space groups by the subgroups of the space groups of the prototype structure. In addition to standard figure-of-merit criteria, the correctness of indexation is confirmed by a clear crystallographic relationship with the prototype structure Indexing the subcell does not require the peak positions to be very precise you can do it even if a zero shift is present. Then use the subcell reflections as a reference to refine the zero shift, correct your data and continue. 19Primitive cubic lattice

very dense set of weak unindexed reflections, which disappear at phase transition, and, hence, also belong to this phase.

Manual search for superstructure seems to be impossible.

Automatic indexing (TREOR90) fails.

Do not waste time, electron diffraction will give straight answer.

24Polymorphic forms of cryolite K3AlF6 the superstructureWe can apply the same concept and separate the reflections into the sublattice ones (brighter) and superlattice ones (less bright).-4 2 0sub 10 0 0ss 2 4 0sub 0 10 0ss 0 0 2sub 0 0 8ss 25Polymorphic forms of cryolite K3AlF6Applying the transformation matrix to the lattice vectors of the substructure and then refining the lattice parameters using the LeBail fit we get (space group I41/a) a 18.8489(2) Å c 33.9827(4) Å V 12073 Å3

Summarizing

Time you spend for indexing pattern with homology method

is proportional to unit cell volume

is inversely proportional to the symmetry of the structure, level of your crystallographic and crystal chemistry expertise and your experience

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